On Gaussian curvature equations in $\mathbb{R}^2$ with prescribed non-positive curvature

Research paper by Huyuan Chen, Feng Zhou, Dong Ye

Indexed on: 16 Oct '18Published on: 16 Oct '18Published in: arXiv - Mathematics - Analysis of PDEs


We study the qualitative properties of the solutions of $(E)$ $ \Delta u +K(x) e^{2u}=0 \quad{\rm in}\;\; \mathbb{R}^2, $ where $K\le 0$ is a H\"{o}lder continuous function in $\mathbb{R}^2$. We find that problem (E) has a sequence of solutions $u_\alpha$ with the asymptotic behavior $\alpha \ln |x|+c_\alpha+o(1)|x|^{-\frac{2\beta}{1+2\beta}}$ at infinity for $\alpha\in(0,\alpha_1(K))$ and any $\beta\in (0,\,\alpha_1(K)-\alpha)$ under the assumption that $\alpha_p(K)>0$ for some $p>1$, where $$\alpha_p(K)=\sup\left\{\alpha>-\infty:\, \int_{\mathbb{R}^2} |K(x)|^p(1+|x|)^{2\alpha p+2(p-1)} dx<+\infty\right\}.$$ Furthermore, we give a more general condition for the existence of solution satisfying $u_\alpha=\alpha\ln|x|+O(1)$ at infinity and we construct new type solution such that the remainder term $u_\alpha-\alpha\ln|x|$ is bounded in $\mathbb{R}^2\setminus B_1(0)$ but does not converge to a constant at infinity or the remainder term $u_\alpha-\alpha\ln|x|$ is unbounded at infinity. To our best knowledge, this is the first model of $K$ such that a solution of $(E)$ with non-uniform behavior at infinity is exhibited.